NOVEL ALGORITHMS FOR IP FAST REROUTE (DRAFT)

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6Figure 3: A pair of node-redundant trees rooted at node D.4.2 Finding Node-Redundant Trees in Linear TimeThis thesis deals with the problem of node-redundant trees (or simply redundant treesin the sequel), a pair of directed spanning trees commonly applied for resilience purposes[Méd99][Méd02][Zha05][Xue02][Cic07]. The rst algorithm with linear complexityfor nding such trees, now widely cited in the literature, was proposed in [Zha05].Denition 4.1. Given an undirected graph G and a root node r, a pair of redundanttrees of graph G rooted at node r is a pair of directed spanning trees of G with edgesdirected towards r. This node is reachable from any other node along both of thetrees, so that the paths in the two trees are node-disjoint.Remark: Obviously, redundant trees can only be found in 2-node-connected graphs,a requirement real networks may not be able to fulll. Lifting this articial assumptionis discussed in the next thesis. In this thesis I always assume that the graph is2-node-connected.Redundant trees can be easily used for resilience. If a single failure shows up, itcan ruin the connectivity towards the root in only one of the trees. Hence, any failurecan be avoided by switching the trac to the tree that remained intact. In the sequeln denotes the number of nodes and e denotes the number of edges of the graph. Apair of redundant trees is depicted on Fig. 3.THESIS 2: [C5][C6][C7] I have found a phenomenon, which makes the algorithm in[Zha05] to fail to nd a correct pair of redundant trees in linear time. I have proposeda centralized algorithm for nding a pair of redundant trees in strictly linear time,which does not suer from this shortcoming. Moreover, I have proposed a distributedalgorithm for nding a pair of redundant trees rooted at each node in linear time.THESIS 2.1: I have found a phenomenon, which makes the algorithm in [Zha05] tofail to give a correct result in linear time. I have shown an example, a graph with aparticular traversal, on which this pathological behavior shows up.
7Figure 4: A graph and a DFS traversal on which no proper address associationexists.Remark: The algorithm in [Zha05] can be modied in order to guarantee correctness,but in this way it necessarily loses linear complexity.The problem of the original solution stems from the fact that this algorithm needsto maintain a partial ordering on the nodes, with insertion and comparison both inO(1) time. Naturally, usual data structures (e.g. arrays, linked lists, binary trees)can not fulll this criteria, but associating platform native numbers to nodes canovercome this problem.Unfortunately, by associating numbers to nodes an address space is created(these are not real addresses, they are only used for comparing), which can run lowwithout proper assignment. Using the graph and the traversal depicted on Fig. 4I have shown that no proper assignment exists; the necessary size of address spacetherefore scales exponentially with the number of nodes of the graph, which can not behandled by the xed size of platform native numbers. Arbitrary precision arithmeticcan be applied, but it does not provide O(1) comparison or insertion.THESIS 2.2: I have proposed Algorithm 1, a correct algorithm for nding redundanttrees with strictly linear, O(e) complexity.Algorithm 1 uses a DFS tree to make an ear decomposition of the graph as theoriginal algorithm does, but in contrast to the algorithm presented in [Zha05], thismethod walks both upward and downward on this tree. In this way, the need ofsupporting a partial ordering is completely eliminated and keeping the complexitylinear becomes possible. Due to space limitations more detailed description can notbe presented here, but it can be found in [C7]. In order to help understanding thefollowing algorithm some notion is discussed here as well.First, DFS number at a given node x is the number of nodes visited before x (thesequence number). Lowpoint number of x is the smaller one from the minimum ofDFS numbers of neighbors and the minimum of lowpoint numbers of node's childrenin the DFS tree. Moreover, ADAG is a special directed spanning graph, usually
Page 2 and 3: 21 IntroductionIP has come a long w
Page 4 and 5: 4Figure 1: Loop in a network with i
Page 8 and 9: 8almost fullling DAG properties. Mo
Page 10 and 11: 10Algorithm 2 Finding the edges (h
Page 13 and 14: THESIS 3.4: I dened heuristics to c
Page 15 and 16: IP Fast ReRoute.Computing the next
Page 17: 17Publication of new results[J] Jou

NOVEL ALGORITHMS FOR IP FAST REROUTE (DRAFT)

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